What experimental observations should lead one to discover the Schrodinger's equation? Not asking what historical observations lead to the discovery. I'm asking what observations do you feel are the best ones and sufficient for the discovery of this equation. So feel free to include observations which were historically a test of the equation instead of a reason for its birth.
Also, by observations, I mean what the universe actually does with its particles. I do not mean observations bound by human technology. So you do not need to precisely set up any specific experiment in the answers. Just state the conclusion drawn from the experiments about the behavior of particles.
 A: This answer does not claim to be the most clever one, but it is more or less the canonical one from the textbooks. Nevertheless, I think my little summary might help you a bit. By the way, nobody knows "what the universe actually does with its particles", we can only describe the universe in terms of observations, and since we as human beings are limited in so many ways, observations are always necessarily incomplete. And there is no guarantee that we will ever be able to extrapolate some "complete theory" from incomplete observations.
Therefore, you won't get a unique and unequivocal path to the Schrödinger equation, just for the same reason why a jigsaw puzzle with half of the parts missing cannot give you a definite impression of the whole image. You might feel very compulsed to think, that the unknown person holding hands with Mr. Kennedy on a fictitious incomplete JFK puzzle was Jaqueline, but this does not rule out the possibility that it was actually Norma Jean, or even someone else. Maybe the picture was taken in private and only very recently been published and exploited by some jigsaw puzzle manufacturer.

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*Observation that electrons behave like a wave (e.g. diffraction at the double slit). Hence, assume that this wave has the form $\psi=exp(i\omega t-i\vec k\cdot \vec r)$

*Observation that wavelength and frequency are related to momentum and energy of the single electron (e.g. accelerated in a cathode ray tube, which is basically nothing but a capacitor) by the de-Broglie relations ($E=h\nu$ and $p=h/\lambda$)

*Conjecture that the most reasonable way to extract energy and momentum (or their expectation values) from a free/plane wave $\psi$ is by applying the derivatives w.r.t. space and time

*Attempt to formulate energy conservation $E=T+V$ for monochromatic matter waves ends up in applying the spatial derivative twice (and multiply some coefficients) to the wave for kinetic energy, applying the first time derivative to the wave for total energy, and finally adding the potential times the wave itself

*due to the simple multiplicative result of the derivative operators for monochromatic waves of any wavelength/frequency/momentum/energy, one might conjecture that an equation that holds true for all plane waves also holds true for all mixtures of those waves (linear superposition principle), which constitutes all possible waves that an electron might follow; this eventually leads to the time-dependent Schrödinger equation $\hat E\psi = \hat T\psi+V\psi$, whereby I have stretched the usual notation a little bit by writing $\hat E$ as an operator (caveat: it is not operating on Hilbert space)

The last point at first might look a bit odd because it means that two superposed waves do not interact (due to linearity). So one might object that electrons are interacting with each other, which seems like a contradiction. But this does not devaluate the concept of matter waves in general, but only shows that the obtained Schrödinger equation is just describing a single particle in an external potential, and not multiple electrons that interact with each other.
In order to let particles interact, some more sophisticated gymnastics is necessary, including the multi-particle Schrödinger equation, the Pauli-principle and Fock space. Although the Pauli exclusion principle is another very important observation, I will refrain from getting into the details here. Because it doesn't end here, once you discover that you need to take special relativity into account, then you need to quantize the EM field as well, asoasf.
A: This question puts the cart in front of the horse.
The mathematical  Balmer  (and other ) series was first used in the Bohr model to give validity for the quantization of angular momentum. The Bohr model succeeded in deriving the series using quantization of angular momentum . In the history it is recorded how the particular non relativistic equation by Schrodinger also gave the series, in a much more reasoned theoretically model.

Despite the difficulties in solving the differential equation for hydrogen (he had sought help from his friend the mathematician Hermann Weyl) Schrödinger showed that his nonrelativistic version of the wave equation produced the correct spectral energies of hydrogen in a paper published in 1926. Schrödinger computed the hydrogen spectral series by treating a hydrogen atom's electron as a wave $Ψ ( x,y,z , t )$ moving in a potential wll   V, created by the proton. This computation accurately reproduced the energy levels of the Bohr model. In a paper, Schrödinger himself explained this equation as follows:


The already ... mentioned psi-function.... is now the means for predicting probability of measurement results. In it is embodied the momentarily attained sum of theoretically based future expectation, somewhat as laid down in a catalog.

Differential equations and their solutions are unlimited.It is the specific observations of the spectrum of the Hydrogen atom that validated the use of the Schrodinger equation and the quantum mechanical theory that developed in consequence as we know it now.
